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A New Horizon in Open Quantum Systems: The Phase-Dependent Jump Rate

Saadat Samadi Dinani

Published Jul 30, 2026• 0 citations
Publisher

Abstract

We propose that the quantum-jump (decay) rate of an open quantum system can depend on the relative phase δ = θ − φ between the system coherence and a reference oscillation, taking the form Γ(δ) = Γ₀ exp(α cos δ) ≈ Γ₀[1 + α cos δ]. The resulting distribution of jump phases is a von Mises distribution whose first Fourier coefficient is M₁ = I₁(α)/I₀(α) ≈ α/2. Standard Lindblad theory employs a constant rate γ₀; this work replaces it with a phase-dependent rate inside the Lindblad dissipator. The operational definitions are explicit: θ = arg ρ_eg is the phase of the system coherence in a fixed reference frame, and φ is the phase of a classical reference oscillation (a drive tone or a phase-lockable bath mode). The relative phase δ is therefore defined only when such a reference exists — pure Fock-state (photon or phonon number) jumps carry no phase information and cannot test the hypothesis. A two-dimensional metric on the torus (θ, φ), g = [[1, −cos δ], [−cos δ, 1]], is offered solely as geometric motivation for the cos δ form. Its Hessian is singular at δ = nπ, but this singularity does not by itself imply an enhanced decay rate, and it cannot select the sign of α. The step from "Legendre failure" to "rate modulation" is an explicit postulate, not a derivation. This work does not claim to solve the measurement problem. A reproducible injection–recovery test with null controls (4000 trajectories per replicate, seed 2026) recovers injected values of α to within statistical error. For the linear law, the estimator α̂ = 2⟨cos δ⟩ is exactly unbiased with standard error √(2/N); for the exponential (von Mises) law, a maximum-likelihood estimator is obtained by solving M₁ = I₁(α)/I₀(α). Under the null (α = 0), the maximum |M₁| observed across 1500 independent replicates is 0.046, with 95th and 99th percentiles of 0.027 and 0.035. Power analysis indicates that a 3σ detection requires N ≈ 2(3/α)² jumps and a 5σ detection requires N ≈ 2(5/α)² jumps. These simulations validate only the statistical estimator: because α was injected by construction, they are not evidence for the physical hypothesis. The manuscript states the platform requirements for a real test (a driven qubit or oscillator with an explicit classical reference phase, measurable system phase from continuous I/Q records, and trajectory-resolved jump detection), the pre-registration requirements, the null controls, and the rival explanations that must be excluded (phase-dependent detection efficiency, ordinary drive-phase dependence, population-dependent rates, and intrinsic heterodyne detector phase response). No experimental result is claimed. The link between the singular metric and the rate modulation is postulated rather than derived; the phases θ and φ must be tied to a concrete Hamiltonian and reference in each physical system; and the sign of α remains a free parameter of the hypothesis. Keywords: open quantum systems; quantum jumps; phase-dependent decay; von Mises distribution; Lindblad equation; quantum trajectories; injection–recovery; null controls; estimator validation

DOI
10.5281/zenodo.23050359

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•Aug 26, 2026•0 citations
We propose that the quantum-jump (decay) rate of an open quantum system can depend on the relative phase δ = θ − φ between the system coherence and a reference oscillation, taking the form Γ(δ) = Γ₀ exp(α cos δ) ≈ Γ₀[1 + α cos δ]. The resulting distribution of jump phases is a von Mises distribution whose first Fourier coefficient is M₁ = I₁(α)/I₀(α) ≈ α/2. Standard Lindblad theory employs a constant rate γ₀; this work replaces it with a phase-dependent rate inside the Lindblad dissipator. The operational definitions are explicit: θ = arg ρ_eg is the phase of the system coherence in a fixed reference frame, and φ is the phase of a classical reference oscillation (a drive tone or a phase-lockable bath mode). The relative phase δ is therefore defined only when such a reference exists — pure Fock-state (photon or phonon number) jumps carry no phase information and cannot test the hypothesis. A two-dimensional metric on the torus (θ, φ), g = [[1, −cos δ], [−cos δ, 1]], is offered solely as geometric motivation for the cos δ form. Its Hessian is singular at δ = nπ, but this singularity does not by itself imply an enhanced decay rate, and it cannot select the sign of α. The step from "Legendre failure" to "rate modulation" is an explicit postulate, not a derivation. This work does not claim to solve the measurement problem. A reproducible injection–recovery test with null controls (4000 trajectories per replicate, seed 2026) recovers injected values of α to within statistical error. For the linear law, the estimator α̂ = 2⟨cos δ⟩ is exactly unbiased with standard error √(2/N); for the exponential (von Mises) law, a maximum-likelihood estimator is obtained by solving M₁ = I₁(α)/I₀(α). Under the null (α = 0), the maximum |M₁| observed across 1500 independent replicates is 0.046, with 95th and 99th percentiles of 0.027 and 0.035. Power analysis indicates that a 3σ detection requires N ≈ 2(3/α)² jumps and a 5σ detection requires N ≈ 2(5/α)² jumps. These simulations validate only the statistical estimator: because α was injected by construction, they are not evidence for the physical hypothesis. The manuscript states the platform requirements for a real test (a driven qubit or oscillator with an explicit classical reference phase, measurable system phase from continuous I/Q records, and trajectory-resolved jump detection), the pre-registration requirements, the null controls, and the rival explanations that must be excluded (phase-dependent detection efficiency, ordinary drive-phase dependence, population-dependent rates, and intrinsic heterodyne detector phase response). No experimental result is claimed. The link between the singular metric and the rate modulation is postulated rather than derived; the phases θ and φ must be tied to a concrete Hamiltonian and reference in each physical system; and the sign of α remains a free parameter of the hypothesis. Keywords: open quantum systems; quantum jumps; phase-dependent decay; von Mises distribution; Lindblad equation; quantum trajectories; injection–recovery; null controls; estimator validation
Abstract Understanding thermalization and irreversibility in open quantum systems requires char-acterizing the competition between coherent unitary dynamics and dissipative environmen-tal interactions. In this work, we present a systematic numerical study of thermalization dynamics in a two-qubit open quantum system governed by a completely positive and trace-preserving Lindblad master equation. The coherent evolution rate is controlled by a real scaling parameter α multiplying the Hamiltonian commutator, while decoherence is indepen-dently controlled by a dephasing rate γ . Although such a scaling of coherent dynamics can be absorbed into a rescaling of time in isolated systems, it becomes physically meaningful in open systems where decoherence introduces an absolute timescale. We investigate how the relative strength of coherent evolution and decoherence influences observable dynamics, purity decay, entropy production, and thermalization times. Using robust local and two-site observables, we show that varying α at fixed γ produces distinct transient trajectories, even though all evolutions converge to the same steady state. We further define an operational thermalization time τ ( γ, α ) based on a purity threshold and map the resulting thermalization-time surface across a broad parameter range. Increasing α generally accelerates oscillatory dynamics and modifies the approach to equilibrium, leading to measurable shifts in thermalization times without altering asymptotic states. These findings provide a clear and physically consistent characterization of how Hamil-tonian strength and decoherence rate jointly determine thermalization behavior in open quantum systems. The framework offers a practical parameterization for numerical mod-eling and experimental studies of coherent–dissipative competition in quantum information processing, quantum control, and open-system dynamics.
We analyze the angular dependence of the purity-loss rate in a restricted class of open quantum systems: a qubit coupled to a thermal bosonic bath through a commuting interaction operator. Within the Born–Markov approximation, we derive the dissipative generator in closed form andestablish four results. First, the restricted induced trace norm of the generator (on the Hermitian traceless subspace) isindependent of the coupling angle, ‖ℒ(𝐷)𝜃 ‖1→1,ℬ0 = 4𝜋𝜂𝑘𝐵 𝑇,and we give a closed-form proof of this fact based on the rotational structure of the coupling family, together with an independent numerical cross-check.Second, the initial purity-loss rate of a pure reference state depends on the coupling angle through the geometric quantity Γ(𝜃) = |⟨𝜓|𝐴(𝜃)|𝜓⟩|2, which for the qubit family 𝐴(𝜃) = cos 𝜃 𝜎𝑥 + sin 𝜃 𝜎𝑧 takes the exact form Γ(𝜃) = (𝑒𝑥 cos 𝜃 + 𝑒𝑧 sin 𝜃)2. Third, the local curvature of Γ(𝜃) at its maximum defines a curvature-derived angular scale 𝜎geo = 1/√2, universal within this coupling family and independent of every physical parameter of the model; the two forms agree to within ∼ 5% at |𝜃 − 𝜇| = 𝜎geo (Appendix E). Fourth, we quantify explicitly the algebraic (non-exponential) decay of the Markov tail integral for the Ohmic–Lorentzian bath considered here, and we show that a slowly time-dependent coupling angle, 𝜃(𝑡), admits a well-defined adiabatic-Markovian generator; we then prove that the resulting purity-loss functional is not proportional, in general, to the trajectory-integrated Frobenius-norm witness of Ref. [1], identifying the specific obstructions (coherent-precession contamination and a linear-versus-quadratic mismatch in Bloch-vector language) responsible, and we pose as an open problem the narrower question of whether any restricted or reformulated version of the witness admits a well-defined relation to the present results. These results provide an exact microscopic benchmark for trajectory-dependent effective decoherence models: the angular dependence originates from the geometry of the system–environment coupling and the evolving state, not from an intrinsic angular preference of the bath, while a full reduction of the phenomenological model of Ref. [1] to this microscopic setting remains open.
Stochastic unravelings provide a useful way to represent open quantum system dynamics in terms of pure state realizations, and have been widely studied both from a fundamental and from a computational point of view. They were initially formulated for Markovian dynamics described by the Gorini-Kossakowski-Sudarshan-Lindblad master equation. However, due to recent technological and experimental development, most physical relevant dynamics present temporal correlations beyond the Markov approximation. Such correlations cause decay rates to turn temporarily negative, thus requiring the generalization of stochastic unravelings from Markovian to non-Markovian scenarios. Indeed, many unraveling techniques have been introduced in this regime, and a comprehensive review of the different jump methods is currently missing. In this work, we provide an overview of widely used quantum jump unraveling techniques for non-Markovian systems and also discuss them in terms of their numerical efficiency, divisibility requirements, Hilbert space extension, and measurement interpretation.
The correct description of the phase variable in quantum mechanics is a question rooted in its earliest formulations. The atom mechanics [1] of Bohr and Sommerfeld—the precursor of modern quantum mechanics—ascribes a central role to action-angle variables. However, Heisenberg's matrix mechanics and Schrödinger's wave mechanics are formulated [2] in terms of the canonical variables representing cartesian coordinates and momenta. Stimulated by this work London [3] attempted a reformulation of these theories in the previously favored action-angle variables. However, this attempt failed due to the difficulty in ascribing quantum operators to the angle variables of classical theory. Despite these difficulties London found [4] an operator representation of the complex exponential of these angle variables. The quantum phase of light made its first appearance in Dirac's classic paper [5] on the quantization of the radiation field. In contrast to modern methods, he constructed the annihilation and creation operators for each field mode from the corresponding amplitude and phase operators. Phase and its quantum nature acquired new significance with the development of lasers in the early sixties: theoretical investigations highlighted significant problems with Dirac's original proposal for the phase operator. A particularly elegant illustration of the difficulty was given by Louisell [6] in 1963. The advent of phase sensitive quantum noise as demonstrated experimentally in the production and detection of squeezed light [7] has created a new wave of interest in the nature of quantum optical phase leading to the discovery of the hermitian optical phase operator [8]. There has followed an explosion of theoretical activity in this area stimulating fresh experimental investigations. In preparing this special issue we have attempted to present the current state of this active and rapidly moving field. We have arranged the papers into what we hope is a coherent representation of the field. The first papers give a historical perspective and overview of current thinking. The two recent experimental investigations which follow are intimately connected to the phase space description of quantum mechanics based on quasi-probability distributions. The representa tion of phase via phase space and its connection with phase-dependent measurements and the phase operator are addressed in the next section. Some more formal considerations pertinent to phase are presented in the following section. Gravitational wave detection and optical communication have motivated the study of the limits of phase noise. Some recent investigations on such optimal phase states are presented. The issue concludes with two papers discussing the significance of phase in light-matter interactions. In concluding we express our gratitude to the authors of the papers in this volume for their efforts in preparing their high quality presentations.
If gravity is information, then the gravitational potential must be an amount of information written at a boundary — and it must be written in a way that does not depend on what is on the other side. That is the informational form of the equivalence principle: a potential is universal, indifferent to the composition of what it acts on. This paper puts that requirement on a quantum processor and measures it.This is a follow-up to Gravity as Approach to a Rate–Resolution Bound (doi:10.5281/zenodo.21808410), and reports the hardware campaign that paper did not contain. In the parity–entanglement programme a crossing write changes a region's bias by ⟨P⟩ → ⟨P⟩|cos 2θ|, so that the amount of record it fixes, δS₂ = −ln|cos 2θ|, has the bias cancelled out of it: it is independent of the state of the region written on, and under Φ = −c²δS₂ it is the Newtonian potential, with θ = v_esc/2c. That independence, and not the mere existence of a correlation between regions, is what separates a gravitational potential from an arbitrary entanglement flow, and it is directly testable. We test it with an estimator that is decay-free by construction — the ratio R_ij(θ) = ⟨P⟩_i(θ)/⟨P⟩_j(θ) of two states carried through identical circuits, in which any depolarising factor cancels identically — and we test it against a deliberately introduced violation. On ibm_kingston, across fifteen independent measurements in two jobs, the sector-exchange write shows no drift: χ² = 12.77 on 15 degrees of freedom, p = 0.62, largest deviation 2.3σ. In the same jobs, and with the same estimator, a control write that is state-dependent by construction is resolved at p = 1.1×10⁻²⁰, and its drift is reproduced by a closed form with no fitted parameter at χ²/dof = 0.52, the largest predicted effect (16.74%) being recovered as 16.59%. The conserved membrane ⟨Z_A Z_B⟩ supplies a second, independent discriminator, dropping by 0.007 in the signal arm and by 0.071 in the control. We also report a measured correction to Eq. (3) of the parent paper: the identity ⟨P⟩ = Tr ρ_A² = e^−S₂ holds in the Gaussian sector but not in the qubit sector, where Tr ρ_A² = (1+⟨P⟩²)/2, confirmed tomographically to 0.3%. Exact density-matrix propagation shows the estimator cancels depolarising, dephasing, bit-flip and coherent write-angle error to machine precision, so a miscalibrated write can manufacture neither the null nor a violation; amplitude damping is the sole surviving channel, at 0.11γ. Every number is re-derived offline from the deposited counts by the accompanying script. Statements are labelled [P] proved, [K] known, [N] numerical or measured, [F] formal.